Modulo Calculator and Order of Operations
Find the nonnegative remainder of integer division, verify the division identity, and see where modulo fits alongside multiplication and division.
Inputs
Required integer from – 1,000,000,000,000 to 1,000,000,000,000.
Required positive integer from 1 to 1,000,000,000,000.
Result
23 mod 5 = 3
23 modulo 5 equals 3.
Calculation steps
How to use this modulo calculator
What this calculator does
This calculator finds the Euclidean modulo of one integer by a positive integer. It reports the nonnegative remainder r in the identity x = q × y + r, where 0 ≤ r < y. It also shows the floor quotient, the reconstructed division identity, and each arithmetic step. It does not evaluate a full typed expression or determine the syntax rules of a specific programming language; those rules can differ, especially for negative values.
When to use it
Use modulo to test divisibility, wrap a counter through a repeating cycle, assign items to repeating groups, or verify a hand calculation involving remainders. It is also useful when checking expressions in code because modulo usually shares precedence with multiplication and division. The Python operator-precedence table, for example, places multiplication, matrix multiplication, division, floor division, and remainder on the same precedence level.
How to calculate
- The calculator opens with a complete demonstration: x = 23 and y = 5. The result and a validated example workbook are available immediately.
- Replace x (dividend) with the integer you want to divide. Replace y (divisor) with a positive integer. Results update as you type.
- Read Remainder r first, then use Integer quotient q, Division identity, Remainder range, and the three calculation steps to verify the result.
- Select Download Excel to create an XLSX workbook from the current validated inputs and outputs. Select Reset to clear the demonstration and all calculated content. After Reset, Excel export stays unavailable until both required fields form a complete valid pair again.
Input guide
x (dividend) is required and accepts a whole number from – 1,000,000,000,000 through 1,000,000,000,000. Enter plain base-10 digits with an optional leading plus or minus sign, such as 23, 0, or – 17. Commas, decimal points, percent signs, scientific notation, and mixed text are rejected rather than silently reinterpreted. Increasing x changes both the quotient and the cycle position represented by the remainder.
y (divisor) is required and accepts a positive whole number from 1 through 1,000,000,000,000, such as 5. Zero is invalid because division by zero has no defined remainder, and a negative divisor is excluded so the displayed range always remains 0 ≤ r < y. A larger divisor expands the possible remainder range; when y is greater than a nonnegative x, the remainder is x itself.
Output guide
Remainder r is the exact nonnegative modulo result. Zero means y divides x evenly. Integer quotient q is the floor of x ÷ y and may be negative. Division identity substitutes the current values into x = q × y + r, while Remainder range confirms the required boundary. The header pills repeat the current dividend, divisor, and remainder for quick scanning. The calculation steps show the quotient, product q × y, and final subtraction x – qy; every value comes from the same canonical calculation used by the workbook.
Worked example
With the startup values x = 23 and y = 5, the floor quotient is q = floor(23 ÷ 5) = 4. Multiplying back gives 4 × 5 = 20. Subtracting gives 23 – 20 = 3, so 23 mod 5 = 3. The identity check is 23 = 4 × 5 + 3, and the range check is 0 ≤ 3 < 5. These exact values appear on first open and in the example XLSX workbook.
Modulo, remainder, and negative numbers
For a positive divisor, Euclidean modulo always returns a value from zero up to, but not including, the divisor. That convention makes repeating cycles predictable: modulo 7 always maps an integer into one of seven positions, 0 through 6. Some programming-language remainder operators behave differently when the dividend is negative. The MDN guide to JavaScript's remainder operator explains that JavaScript's percent operator follows the dividend's sign and shows how to normalize it to modulo. This calculator instead uses floor division with a positive divisor, so – 13 mod 5 is 2 because – 13 = ( – 3 × 5) + 2.
Where modulo sits in the order of operations
In common programming grammars, the percent operator is grouped with multiplication and division. Operations at that level are normally evaluated from left to right, after parentheses and exponentiation but before addition and subtraction. Thus, an expression such as 2 × 3 % 4 is handled as (2 × 3) % 4, producing 2. The language specification remains the final authority: the ECMAScript multiplicative-operator specification defines the grammar used by JavaScript. In ordinary mathematical writing, parentheses are the safest way to remove any ambiguity about whether “mod n” describes one operation or the arithmetic environment for a larger expression.